2. 1
1: Executive Summary
The objective of the project is to determine the best overall engine cycles for two
different vehicles at various flight conditions. The two vehicles are a commercial airliner and a
long range missile.
For the commercial airliner, the optimized turbofan design provides the best overall
performance. At ground roll, the engine provides a specific thrust of 2.743 kN*s/kg and TSFC
(thrust specific fuel consumption) of 0.015 kg/kN*s. At high altitude, the engine provides a
specific thrust of 0.905 kN*s/kg and TSFC of 0.022 kg/kN*s. For the long range missile, the
turbojet provides the best overall performance. At low altitude, the engine provides a max
specific thrust of 1.71 kN*s/kg and fuel optimized TSFC of 0.02 kg/kN*s. At the high altitude
condition, the engine provides a max specific thrust of 1.3881 kN*s/kg and fuel optimized TSFC
of 0.0241 kg/kN*s.
The optimized turbofan design has the following specifications: Prc=44, Prf=1.34,
Bypass=1, b=0.12, combined nozzle. The optimized turbojet has the following specifications:
Prc=40, b=0.12.
2: Introduction
In this project we were challenged to create three models for four different engine cycles
and assign two vehicles the most appropriate engine. Each vehicle operates at two different flight
conditions. After each vehicle was assigned an engine, the engine parameters were optimized for
the cruise flight condition of the vehicle. The engines were optimized with respect to fuel
efficiency, while still maintaining the required thrust at both conditions. The full data for each
optimized engine was then tabulated.
3. 2
The modeling approach was to separately model the three main engine types. The three
main types of engines are ramjets, turbojets, and turbofans. We created an individual engine
cycle for both ramjets and turbojets. The turbofan had two distinct cycles because it has more
parameters to vary than the other two engines. Each group member was assigned an engine
except for the turbofan, which was assigned to two group members due to its complexity. The
equations were initially developed with Mathematica or by hand. We used cycle analysis to
develop the equations, this is further explained in section 3.1.1. These equations were then
distributed amongst the group as a way to verify them. These equations were revised in some
cases. These versions are found in Appendix A. The development of the equations can be found
in Appendix B. Then using these equations three different Matlab functions were created. The
function inputs included the inputs that varied with each condition and the engine parameters that
were fixed and variable. The turbofan’s functions fixed parameters were varied in order to create
two different engine cycles.
Initial analysis, the purpose of this analysis was to start sorting which engines would be
remotely adequate for each vehicle. The exact parameters (b, Prc, Prf, etc) for each engine at first
were largely chosen at random. After this analysis was done the competing engines for each
vehicle became evident. The two competing engines for each vehicle ( 4 in total since 2 vehicles)
were then directly compared using plots overlaid containing two sets of data. The engine most
appropriate for each vehicle was then chosen, with efficiency in mind.
Now that each vehicle had a specific engine assigned, we focused on achieving the best
fuel efficiency while still providing the required thrust for both operating conditions. In this
section the different engine parameters fixed and variable were plotted against TSFC and SFC,
the remaining parameters were kept constant. This allowed the most fuel efficient, and still
4. 3
meeting the req. thrust, (we ran into issues meeting this requirement), parameter to be chosen.
After this the full data output (Ti, Pi, wi, etc) for the optimized appropriate engine and the
directly competing engine was tabulated. This is intended to provide a final proof of the choice,
also to highlight the differences in fuel economy and efficiencies.
3: Results and Discussion
The results are displayed to justify the decisions that were taken to narrow down which
engine with which parameters is the best for the two vehicles. This process of elimination was
done through specific comparison of cycles. The first section eliminates cycles from vehicles if
they are not able to meet the required thrust conditions. This leaves narrows it down to two
cycles per vehicle from the original four. Then these two cycles per vehicle are pitted against
each other. After that most appropriate cycle is chosen per vehicle, this cycle’s parameters are
now optimized for the vehicle. After this optimization has been performed the complete data for
the cycle is displayed along with the complete data for the rejected cycle (this is meant to
highlight the differences of each cycle)
Table 1: Ramjet with Varied Flight Conditions
As described by Table 1, the Ramjet, when flying at different flight parameters, a few of
the parameters being Mach 0, 0.85, 0.80, 2.4, and their corresponding altitudes of 0, 9.14, 6.25
and 15.2 km, these conditions were directly associated with the airliner and long range missile
vehicle performance charts that are displayed in Table 1 of the project description. The ramjet
was able to meet both req. thrusts for the missile. With the airliner flight parameters, the ramjet
5. 4
can’t achieve its required thrust at either flight condition. Also, table 1 shows that at a higher
altitude and Mach number, the TSFC decreases allowing the ramjet to consume less fuel per
thrust, then the lower altitude, and Mach number, conditions. The ramjet is meant to function at
high Mach values.
Table 2: Turbojet with Varied Flight Conditions
The turbojet is the most robust engine cycle. It is able to perform relatively well at almost
all of the flight conditions. It was not able to meet the required thrust for the airliner ground roll.
This is due to the limits of the compressors ability to compress the air before it is combusted.
The turbojet operates best within the range of Mach 0.8 to 2.5. As the Mach number increases
past around 1 its TSFC will increase. At around Mach 2.8 this increase will become sharp and
make the turbojet unusable.
Table 3: Turbofan Design 1 with Varied Flight Conditions
The turbofan at this configuration with the maximum fuel-air ratio can produce the
required specific thrust for the airliner at high altitude and the missile at low altitude. However,
this turbofan fails to meet the required specific thrust of the vehicles at their respective low and
high alt. flight conditions. As such, this engine configuration should not be used for either of the
vehicles.
6. 5
Table 4: Turbofan Design 2 with Varied Flight Conditions
The turbofan is an engine designed to have high propulsive efficiencies which is achieved
by having a high bypass ratio. For this engine cycle a high bypass ratio is chosen to decreases the
TSFC so that the propulsive efficiencies are high. Furthermore, the second turbofan cycle as seen
in table 4 meets all of the required specific thrust conditions except for the case of the missile at
high altitude. The reason for this failure is the inability of this turbofan cycle to operate at high
Mach numbers with the bypass ratio being so high. The best performance for this turbofan cycle
is for an airliner achieving takeoff as the specific thrust in this case is the highest as such is need
in a static thrust condition while also having the lowest TSFC.
Section 3: Results and Discussion
Section 3.1: The comparison of the Ramjet vs Turbojet
For vehicles that perform in high speed flight conditions, typically the two main vehicle
engine that are used are the ramjet and turbojet. The ramjet is a relatively simple engine that
scoops in air, injects fuel into the air, and the lit fuel-air mixture provides the thrust for the
ramjet. The turbojet, on the other hand, is a drastically more complicated engine. While able to
operate in a wide operating envelope, including supersonic conditions, the turbojet requires that
the air that get sucked into the engine gets compressed before it gets lit in the burner. Because of
this, the turbojet is not a solid-state engine, whereas the ramjet is. Due to this mechanical
addition in the engine, its overall performance is restricted to much lower Mach conditions,
where the ramjet excels in higher Mach ranges. The comparison of the two engine designs are
shown in Figure 3.1.1.
7. 6
Figure 3.1.1: Ramjet vs Turbojet Engine
Because the ramjet and turbojet have different engine characteristics, their performance
properties can be modelled and graphed to determine the best flight conditions and ranges for
each engine. These performance properties are designed by applying cycle analysis on each
engine. Cycle analysis is the breaking up of the engine into components, and using design
parameters, determining the change of effects in each component. An example of this is, one
component might only increase in pressure, whereas another component might add mass to the
airflow mass running through the engine.
By carefully working through each component of the engines, abiding by internal
restrictions, such as maximum temperature ranges, restrictions of how much compression can be
achieved, etc., the two primary performance characteristics, the engine specific thrust, otherwise
known as SFC, which is defined as the thrust/mass-air-flow, gives the specific thrust the engine
can achieve. The second performance characteristic to take into effect is the thrust specific fuel
consumption, or otherwise known as TSFC, which is defined as ratio of thrust and fuel flow rate.
This parameter is directly proportional to how much fuel is used for any given amount of
thrust. One of a designers criteria is to maximize fuel efficiency, so minimizing the TSFC is one
of the design parameters that must be met. There are additives to the engines, such as an
afterburner to a turbojet, that massively increases how much fuel is being used, but the return for
such a device is an increase in speed. Such external design parameters much be taken into
8. 7
consideration when the operating criteria of the vehicle is given. In certain applications, an
increase of speed by using a significantly higher amount of fuel is one of the design parameters
that must be met. An example of this is if a military aircraft needs to increase in speed to get
away from an enemy, or to get to an objective faster, the pilot would rather achieve maximum
thrust instead of trying to conserve fuel. But in general, for any engine, when designing an
engine, the specific thrust is typically maximized while the fuel consumption rate is minimized.
For the ramjet and turbojet, the performance variations from low to high mach values are given
in Figure 3.1.2.
Figure 3.1.2: High Altitude Rocket Performance Comparisons
There are a variety of comparisons that need to be made when comparing a ramjet and
turbojet at high altitude conditions. First and foremost, the designs must meet the required thrust
parameters for the flight condition. If one of the engines can’t meet this requirement, no matter
how fuel efficient the engine is, the vehicle cannot operate in this condition. As shown in Figure
3.1.2, up until really high Mach ranges, the turbojet has higher specific thrust outputs with a
much lower fuel consumption rate. The graphs show though that after a while, around Mach 3,
the turbojet stars to fall off dramatically, falling below the required thrust condition, while the
ramjet can still maintain its required thrust at a much lower fuel consumption rate. This comes to
9. 8
show that for a high altitude missile, if the missile is being designed for really high Mach
conditions, it should be designed with a ramjet, otherwise a turbojet should be used for the
design.
Figure 3.1.3: Low Altitude Rocket Performance Comparisons
Another major operating condition for the ramjet and turbojet is being able to operate in
low altitude conditions. Due to changes in ambient temperature and pressure, the required SFC
ends up being significantly less. The performance envelopes for the ramjet and the turbojet then
can be plotted against each other to determine which engine performs at different flight speeds.
As shown by Figure 3.1.3, if the vehicle needs to fly at a lower speed, the turbojet ends up
having a larger specific thrust with lower fuel consumption. But if it needs to operate in
conditions higher than around Mach 2.5, the ramjet is a much better choice as it would have a
higher specific thrust and lower fuel consumption rate. This correlates with the high altitude
performance comparison charts (Figure 3.1.2), that the higher the speeds, the better the ramjet
performs, but if operating at lower speeds, the turbojet is the better choice of the two engines.
10. 9
Section 3.2: The comparison of Turbofan Design 1 vs Turbofan Design 2
Since there are two engine cycles for the turbofan, with dramatically different fixed
parameters a comparison between them is needed. The comparison will show how each engine
cycle performs at varying flight conditions and demonstrate that one cycle is better than the
other. The two cycles design parameters are given in Table 3 (High Altitude Subsonic Cruise -
HASC) and Table 4 (Ground Roll)
11. 10
Figure 3.2.1: Specific Thrust vs Mach Number Comparison
Comparing the specific thrust curves with varying Mach number as shown in Figure 3.2.1. In
comparison the high bypass cycle (listed as Ground Roll) can be seen to have the higher specific
thrust at low Mach numbers (M< 0.4) such that they fall into the incompressible flow regime;
while the HASC cycle has higher values for specific thrust at higher Mach numbers (0.4 < M <
1). Another, analysis concluded from Plot 3 is that the ground roll cycle shows that as Mach
number increases beyond Mach 0.7 the specific thrust goes negative whereas the HASC cycle
can be seen to exhibit a trend toward a slow decline in the specific thrust values. Thus from the
comparisons in Figure 3.2.1 the better cycle is the HASC cycle since the ground roll cycle can
only achieve a maximum Mach number of about 0.7 when a typical commercial airliner has a
cruising Mach number of roughly 0.85.
12. 11
Figure 3.2.2: TSFC vs Mach Number Comparison
The comparison of TSFC with varying Mach number for ground roll and HASC cycles is
shown in Figure 3.2.2. By comparing the ground roll and HASC cycles using TSFC curves
shows where each cycle is the most efficient. The first observation is that for the ground roll the
TSFC curve has its lowest point at a Mach number of 0 whereas the HASC cycle has its highest
point at a Mach number of 0. Further examination of Figure 3.2.2 show that the ground roll cycle
when it approaches a Mach number of roughly 0.8 the TSFC curve shoots off toward infinity.
Thus a final observation of ground roll and HASC is that the HASC cycle can achieve higher
Mach numbers on its TSFC curve than the ground roll cycle can.
With the observations from Figure 3.2.1 and Figure 3.2.2 a conclusion can be reached
that the HASC cycle is the better option. This conclusion can be reached by simply due to the
fact that the high bypass cycle is unable to reach the req. thrust at condition 2.
13. 12
Section 3.3: Optimized Turbojet Design Analysis
The parameters of the Turbojet that can be optimized for efficiency are the bleed ratio,
compressor ratio, and the fuel consumption of the afterburner. The bleed ratio is largely
independent of both the afterburner fuel and the compressor ratio, so that will be optimized first.
The ambient conditions for all these plots is high alt. supersonic cruise. We make the assumption
that the missile will spend most of its flight at this flight condition. The engine must still meet
the req. thrust for both conditions.
Figure 3.3.1: Bleed Optimization
In these plots the bleed ratio was varied and the fuel for the afterburner and main burner
was consistently kept at their max all other parameters were constant and equal to the flight
conditions at high alt. supersonic cruise. It is evident that the max bleed of 0.12 is the optimized
condition for both TSFC and SFC. TSFC should be minimized in order to maximize fuel
economy. This result is also be applicable to turbofans.
14. 13
Figure 3.3.2: Afterburner Fuel Ratio Optimization
In these plots the afterburner fuel ratio was varied the fuel for the main burner was kept at its
max, all other conditions were constant. The afterburner is a tradeoff, you are able to achieve a
higher specific thrust at the cost of higher TSFC, which means more fuel consumption per thrust.
If it is possible to reach the desired specific thrust then the afterburner fuel ratio should be zero.
The afterburner is much more inefficient than the main burner, so fuel ratio for the main burner
will be kept at its max. If the missile requires the extra thrust the afterburner should be used as a
last resort.
15. 14
Figure 3.3.3: Compressor Ratio Optimization
The red line in the plot indicates the required thrust condition. In these plots the
compressor ratio was varied the fuel to the main burner was kept at its max and the afterburner
fuel ratio was set to 0. The compressor operates at the highest fuel ratio that meets the required
thrust condition. The compressor ratio (Prc) is most efficient at higher values but it is not able to
meet the req. thrust hence the highest Prc that meets the req. condition is optimum, Prc=40.
The above plots indicate that bleed ratio should be set to max, 0.12, the afterburner fuel
ratio should be 0 and the compressor ratio should be 40. These design parameters were tested at
the low alt. condition, as the vehicle must operate at both conditions, and the specific thrust was
0.151 (kN*s/kg) above the required thrust. The thrust specific fuel consumption of the optimized
turbojet at the high alt. cruise is 37% better than the original turbojet. The detailed data is
provided in section 5.
16. 15
Section 3.4: Optimized Turbofan Design Analysis
Since the vehicle will spend most of its time at cruise the fuel efficiency optimization was
created at the cruise condition. Thus, an optimization is desired for the HASC cycle to find the
optimal values for the fan pressure ratio, the compressor pressure ratio, and the bypass ratio by
computing specific thrust and TSFC curves.
Figure 3.4.1: Fan Pressure Ratio Optimization
For the optimization of the fan pressure ratio Figure 3.4.1 will be used. To optimize the
specific thrust the maximum specific thrust (red circle) must be found such that it is above the
required specific thrust line (black line). To optimize the TSFC the minimum TSFC value (red
circle) is desired. The optimum value that gives the lowest TSFC and meets the req. thrust is
1.34.
17. 16
Figure 3.4.2: Compressor Pressure Ratio Optimization
For the optimization of the compressor pressure ratio Figure 3.4.2 will be used. To
optimize the specific thrust the maximum specific thrust (red circle) must be found such that it is
above the required specific thrust line (black line). To optimize the TSFC the minimum TSFC
value (red circle) must be found. Therefore, finding the value for the optimal compressor
pressure ratio to be at the point where the minimum TSFC value is found to be 44.
Figure 3.4.3: Bypass Ratio Optimization
The optimal bypass ratio is found from using Figure 3.4.3. For finding the optimal bypass
ratio from the maximum specific thrust (red circle) and minimum TSFC (red circle) values the
optimal bypass ratio must be found above the required specific thrust line(black line) The
18. 17
optimal bypass ratio is found to be 1 for the maximum specific thrust. The optimal bypass ratio is
also found to be 1 for the minimum TSFC value. The issue with this result is that a bypass of one
engine does not produce the thrust req. for the ground roll condition. For the purpose of this
project we choose to set the optimized bypass at 1 in order to show what the best possible engine
would be able to produce at the high alt. condition. A project focusing on optimization would
need to take this discrepancy into account.
In conclusion the optimized values for the fan pressure ratio, the compressor pressure
ratio, and the bypass ratio come out to be 1.34 for the fan, 44 for the compressor, and 1 for the
bypass ratio, keeping in mind the ground roll condition is not achievable.
19. 18
Section 3.5: Optimized Design Tables
The optimized values found for the turbojet and the turbofan were used to find the engine
parameters in Figure 3.5.2 and Figure 3.5.4. Figure 3.5.1 and 3.5.3 show the engine parameters
for the engine cycles that either failed to meet the requirements of the flight conditions or were
not the best choices.
Figure 3.5.1: Ramjet
Figure 3.5.2: Optimized Turbojet
The difference in fuel consumption and efficiencies between the two engines is striking, an
overall efficiency of 28% vs 65.5% is more than significant and justifies our engine choice.
20. 19
Figure 3.5.3 Ground Roll Turbofan Cycle
A table for the above engine at high alt. is not possible due to the Mach not being achievable.
Figure 3.5.4 Optimized Turbofan at HASC
The above table cannot be directly compared due to the rejected turbofan not being able to
achieve this flight condition.
21. 20
Section 4: Conclusions and Appendices
One of the most important items when designing any type of aircraft is which engines to
use. The aircraft cannot fly without engines, and knowing which engines perform well in
different situations is absolutely vital. An Airliner that is trying to achieve the maximum flight
distance possible is not going to have design parameters around how fast it can go, but instead,
how effectively it can conserve fuel in order to achieve a maximum flight range. On the other
hand, a missile is aiming for speed, so it can catch up to its target, or quickly, and efficiently,
strike a target. Because of this, fuel conservation is less important than the attempt to reach a
maximum thrust.
In this design analysis, each flight range was analyzed in a variety of conditions, the
primary conditions being different altitudes (which had corresponding pressure and temperature
changes), and different speeds for each vehicle. As shown in Figure 3.2.1, for low Mach speeds
(0-0.85), the Turbofan is able to produce a large amount of specific thrust (SFC) for a relatively
low amount of fuel (as shown in Figure 3.2.2). For performance in high Mach conditions, on the
other hand, the turbojet produces a much larger SFC then the ramjet, from Mach 0-2.5. This is
shown in Figure 3.1.2, and for the range of Mach 0-2.5, the turbojet has a much lower TSFC as
shown in Figure 3.1.3. But if the design parameters call for high Mach ranges of M > 2.5, the
turbojet falls off in SFC and TSFC, and the ramjet ends up being more efficient.
Once a given flight range is designated, each engine can be optimized to achieve its
maximum SFC and TSFC. By varying a range of values, such as bleed air, compression ratios,
etc. the results can be plotted and the best performance ranges can be determined. An example of
this is varying the bleed air. As shown in Figure 3.3.1, the higher the bleed air, the lower the
22. 21
TSFC. So the engine would need to perform at a higher bleed ratio in order to minimize fuel
consumption. Another example is the bypass air ratio. As shown in Figure 3.4.1, as the pressure
ratio increases, there is an area where the TSFC drops, and the optimal flight condition is where
the plot is as a minimum.
The overall result of this analysis is that when designing an engine, the first parameter
that is considered is if the vehicle is going for fuel efficiency or speed. Based off of that
parameter choice, an appropriate engine can be chosen and then optimized. Because of the
specific requirement asked for, there isn’t a single “best engine” choice for a given aircraft. The
best engine design is the design that fits the required parameters for the initial problem that is
given, and then that engine can be optimized to provide the best appropriate engine for the
aircraft design. For our vehicles and test cases the best engines are listed in section 3.5 along
with the full set of parameters and data.
25. Appendix A - Ramjet Equations
Section 1: Diffusor Equations
Cp
Γ
Γ 1
R
MW
; This varies per section
U M Γ R Ta ;
T01 Ta 1
Γ 1
2
M2
;
rd If M 1 rd 1, M 1 rd 1 0.075 M 1 1.35
;
P01 rd. Pa
T01
Ta
Γ
Γ 1
Ηd;
Section 2: Burner Equations
fmax
1
T01
Tmax
Hrf
cp
1
;
T04
Cp T01 f Hrf Ηb
1 f Cp
Pf .2 P01 550
Wp
Pf .1 Pf .2
Ρ
Section 3: Nozzle Equations
Pe
P01
1
u2
2 Cp Ta
Cp
R
;
P0 e
P01 Pe
Pa
;
Printed by Wolfram Mathematica Student Edition
26. Me
2
Γ 1
1
Γ 1
2
M2
P0 e
P01
Pa
Pe
Γ 1
Γ
1 ;
Ue Me
Γ R T04
1
Γ 1
2
Me
2
;
Te
2 Cp T04 Ue
2
2 Cp
;
SFC
1 f Ue U
1000
;
TSFC
f
SFC
;
Section 4: Efficiencies
Ηp
2 U
Ue
1
U2
Ue
;
Ηth
1 f Ue
2
2
U2
2
Hrf f
;
Ηo 2 Ηth
U
Ue
1
U
Ue
;
2 Appendix A Ramjet Equations.nb
Printed by Wolfram Mathematica Student Edition
37. Appendix B - Ramjet Equation
Development
Diffusor
All Equations were standard.
Burner
For the Ramjet Burner, the limiting values are based off of the afterburner restrictions due to the fact the
air from the burner doesn’t have to run through a turbine (it’s compressed air lit with flame holders), so
the burner can have a higher Tmax at 2200K.
To solve for the effective T04, this can be achieved by using the fuel air mixture equation:
fmax
1
T01
Tmax
Hrf
cp
1
;
If we re-arrange the equation to solve for T04, and add in the burner efficiency:
T04
Cp T01 f Hrf Ηb
1 f Cp
Also, for the burner, we calcuate the Cp using the Cp equation, and Γ = 1.32
Cp,b
Γb
Γb 1
8314
28.8
;
The fuel pump work was also calculated using the definition of Work for the fuel pump.
Pf .2 P01 550 Pf.2 given design parameter
Wp
Pf .1 Pf .2
Ρ
Nozzle
For the exit pressure, the exit Cp for the nozzle had to be calculated using Γn=1.35):
Printed by Wolfram Mathematica Student Edition
38. Cp,n
Γn
Γn 1
8314
28.8
;
Using this Cpvalue, the diffusor exit pressure P01), ambient temperature Ta), and the molecular weight
for air (R), the corresponding exit pressure was calculated as follows:
Pe
P01
1
u2
2 Cp Ta
Cp
R
;
As stagnation pressure is not standard throughout the engine (as an Ideal Engine would have
P01 P0 a), the change of stagnation at the exit had to be determied as follows:
P0 e
Pe
P01
Pa
By re-arranging the formula, the stagnation exit pressure can be solved for:
P0 e
P01 Pe
Pa
;
The exit velociity is taken in terms of stagnation temperature, so it is then defined as:
Ue Me
Γ R T04
1
Γ 1
2
Me
2
Finally, the performance variables SFC and TSFC are standardized as the following equations and
didn’t have to be derived:
SFC
1 f Ue U
1000
;
TSFC
f
SFC
;
Efficiencies
The performance efficiencies are defined as follows:
Ηp
2 U
Ue
1
U2
Ue
;
Ηth
1 f Ue
2
2
U2
2
Hrf f
;
Ηo 2 Ηth
U
Ue
1
U
Ue
;
One of the efficiencies that is overlooked is the propeller efficiency Ηpr . As the ramjet doesn’t have a
propeller, this efficiency is overlooked adn not included with the overall efficiency.
2 Untitled-2
Printed by Wolfram Mathematica Student Edition
39. One of the efficiencies that is overlooked is the propeller efficiency Ηpr . As the ramjet doesn’t have a
propeller, this efficiency is overlooked adn not included with the overall efficiency.
Untitled-2 3
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40. Turbojet Equation Development
Most Equations were general equations, the ones shown here were developed specifically for this project
In[10]:= Clear;
Diffuser
All Equations were standard
Fan
Turbojet Lacks fan hence air is just passed through
Compressor
In[11]:= P03 Prc P02;
From Isentropic relations
In[12]:= T03 s T02
P03
P02
Γc 1
Γc
;
In[13]:= T03 s T02 Prc
Γc 1
Γc ;
Definition
Ηc
T03 s T02
T03 T02
; Polytropic effciency Calulated with reg. equation
In[15]:= T03 s T02 T02 Prc
Γc 1
Γc 1 ;
In[16]:= T03 T02
1
Ηc
T03 s T02 ;
In[17]:= T03 T02
1
Ηc
T02 Prc
Γc 1
Γc 1 ;
In[18]:= T03 T02 1
1
Ηc
Prc
Γc 1
Γc 1 ;
Compressor Work
In[1]:= h03 s h02 Wc;
Printed by Wolfram Mathematica Student Edition
41. C.P.G
Cpc T03 T02 Wc;
From above development
Cpc
1
Ηc
T02 Prc
Γc 1
Γc 1 Wc
Wc
Cpc T02
Ηc
Prc
Γc 1
Γc 1 ;
Burner
mb h04 mc h03 m Qr Ηb
C.P.G
mb Cpb T04 mc Cpc T03 m Qr Ηb
Divide all by ma dot
In[3]:= 1 b Cpb T04 1 b Cpc T03 Qr Ηb
Solve for T04
T04
1 b T03 Cpc Qr Ηb
Cpb 1 b
Max fuel ratio- from above
1 max b Cpb Tmax 1 b Cpc T03 max Qr Ηb
max Cpb Tmax Qr Ηb 1 b Cpc T03 1 b Cpb Tmax
max Qr Ηb Cpb Tmax 1 b Cpb Tmax Cpc T03
max
1 b Cpb Tmax Cpc T03
Qr Ηb Cpb Tmax
;
Turbine Exhaust
The work the turbine extracts from the gas regardless off effciences is equal to Wc
mT h04 h05.1 Wc ma
Divide all by ma dot
1 b h04 h05.1 Wc
C.P.G
1 f b Cpt T04 T05.1 Wc
2 Equ Develop.nb
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42. T05.1 T04
Wc
Cpt 1 b
Development of Pressure exhaust very similiar to compressor Temp.
P05.1 s
P04
T05.1
T04
Γt
Γt 1
P04 P05.1 s P04 1
T05.1
T04
Γt
Γt 1
Definition
Ηt
P05.1 s P04
P5.1 P04
P04 P05.1
1
Ηt
P04 1
T05.1
T04
Γt
Γt 1
P05.1 P04 1
1
Ηt
1
T05.1
T04
Γt
Γt 1
;
Development of Turbine work
mT h04 h05.1 s Wt mT
Cpt T04 T05.1 s Wt
cpt T04 1
T05.1 s
T04
Wt
Cpt T04
Ηt
1
T05.1 s
T04
Wt
Bleed Mixer
mtm h05 tm mT h05.1 mb h03
mtm mT mb
1 h05 tm 1 b h05.1 b h03
C.P.G
1 Cptm T05tm 1 b Cpt T05.1 Cpc b T03
T05tm
1 b Cpt T05.1 Cpc b T03
1 Cptm
Equ Develop.nb 3
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43. The isentropic result mixing of two gasses flowing to a third is
P03
P02
P01
P02
1
Σ 1 T03
T02
cp
R T02
T01
cp
R
1
Σ 1
cp
Γtm
Γtm 1
;
1
Σ 1
b
1
; P03 P05tm;
T05tm T03; P01 P03; T01 T03; T05.1 T02; P05.1 P03
P05tm P05.1
P03
P05.1
b
1 T05.1
T03
b
1
Γtm
Γtm 1 T05tm
T05.1
Γtm
Γtm 1
;
Afterburner
mab h06 mb h05tm m ab Qr Ηab
CPG
mb Cpab T06 mc Cptm T05tm m ab Qr Ηab
Divide by mass of air
1 ab Cpab T06 1 Cptm T05tm ab Qr Ηab
T06
Cpt T05tm 1 ab Qr Ηab
Cpab 1 ab
;
Max Fuel air ratio form above
1 abmax Cpab Tmaxab 1 Cptm T05tm maxab Qr Ηab
Solve for abmax
abmax
1 Cpab Tmaxab Cpt T05tm
Qr Ηab Cpab Tmaxab
;
Nozzle
Pe Pa;
Tes
T06
Pe
P06
Γn 1
Γn
Tes T06 T06 1
Pe
P06
Γn 1
Γn
Te T06 ΗnT06 1
Pe
P06
Γn 1
Γn
4 Equ Develop.nb
Printed by Wolfram Mathematica Student Edition
44. Te T06 1 Ηn 1
Pe
P06
Γn 1
Γn
;
Ue2
2
h07 h7
Ue 2 h07 h7 ; h07 h06
Ue 2 Cpn T06 T7
Cpn Rn
Γn
Γn 1
Ue 2 Rn
Γn
Γn 1
T06 T7 ; T7 Te
Ue 2 Rn
Γn
Γn 1
T06 Te
From Above
Te T06 ΗnT06 1
Pe
P06
Γn 1
Γn
Ue 2 Rn
Γn
Γn 1
ΗnT06 1
Pe
P06
Γn 1
Γn
Specific Thrust
cs
ux Ρ u.n a me ue ma u
Τ me ue ma u Pe Pa Ae
Neglect Pressure difference
Τ me ue ma u
Τ
ma dot
1 ab Ue U
Equ Develop.nb 5
Printed by Wolfram Mathematica Student Edition
45. Turbofan
Diffuser Exhaust Temperature
ma m1
ma hoa m1 ho1
hoa ho1
ho1 ha
u2
2
To1 Ta
u2
2 cpd
Ta 1
Γd 1
2
M2
Diffuser Exhaust Pressure
Po1
Pa
To1s
Ta
Γd
Γd 1
Po1 Pa 1 Ηd
Γd 1
2
M2
Γd
Γd 1
Fan Exhaust Temperature
To2
To1
Po2s
Po1
Γf 1
Γf
To2 To1 1
1
Ηf
Prf
Γf 1
Γf 1
Compressor Exhaust Temperature
m2 m3
m2 h02 Wc m3 ho3
wc cpc To3 To2
To3 To2
1
Ηc
To3s To2
To3s
To2
Po3s
Po2
Γc 1
Γc
To3 To2 1
1
Ηc
Prc
Γc 1
Γc 1
Burner Exhaust Temperature
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46. m3 mf m4
m4 ma 1 f
Ηb f hr m3 ho3 m4 ho4
Ηb f hr 1 f ho4 ho3
f
To4
To3
1
Ηb hr
cp To3
To4
To3
To4
1
1 f b
f hr
cpb
To3
Turbine Exhaust Temperature
m4 m5
m4 ho4 m5 ho5 .1 WT
WT Wc
To5 To4
wc
1 f cpt
Turbine Exhaust Pressure
Po5 .1
Po4
To5s
To4
Γt
Γt 1
Po5 .1 Po4 1
1
Ηt
1
To5 .1
To4
Γt
Γt 1
Afterburner Exhaust Temperature
m5 mfab m6
ma 1 f fab m6
fab
mfab
ma
m5 ho5 Ηab mfab hr m6 ho6
To6 1 f To5 .1
Ηab fab hr
cpab
1
1 f fab
Core Nozzle Exhaust Temperature
m6 me ma 1 f
Ηn
ho6 hoe
ho6 hoes
Te To6 1 Ηn 1
Tes
To6
Te To6 1 Ηn 1
Pe
Po6
Γn 1
Γn
Core Nozzle Exhaust Velocity
2 Turbofan Appendix B.nb
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47. m6 ho6 me hoe me he
ue
2
2
ue 2 cpn To6 1
Te
To6
1
2
Power per Airflow Rate
m4 ho4 Wc Wf m4 ho5
m4 cpt To4 To5 Wc Wf
wc wmt cpc To3 To1
wf wft Β cpf To2 Toa
Turbofan Appendix B.nb 3
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